Summary: TAYLOR POLYNOMIALS FOR NABLA DYNAMIC EQUATIONS
ON TIME SCALES
DOUGLAS R. ANDERSON
Abstract. We are concerned with the representation of polynomials for nabla
dynamic equations on time scales. Once established, these polynomials will be used
to derive Taylor's Formula for dynamic functions. Several examples are given for
special time scales such as Z, R, and qZ for q > 1. These polynomials will be
related to those given for delta dynamic equations.
1. Preliminaries on Time Scales
The following definitions can be found in Agarwal and Bohner [1], Atici and Gu-
seinov [2], and Bohner and Peterson [3]; these are based on the notions first introduced
by Hilger in his Ph.D. thesis [4]. A time scale T is any nonempty closed subset of R.
It follows that the jump operators , : T T
(t) = inf{s T : s > t} and (t) = sup{s T : s < t}
(supplemented by inf := sup T and sup := inf T) are well defined. The point t T
is left-dense, left-scattered, right-dense, right-scattered if (t) = t, (t) < t, (t) = t,
(t) > t, respectively. If T has a right-scattered minimum m, define T := T - {m};
otherwise, set T = T. If T has a left-scattered maximum M, define T
:= T - {M};
otherwise, set T